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**Etendue** or **étendue** (/ˌeɪtɒnˈduː/; French pronunciation: [etɑ̃dy]) is a property of light in an optical system, which characterizes how "spread out" the light is in area and angle. It corresponds to the beam parameter product (BPP) in Gaussian beam optics. Other names for etendue include **acceptance**, **throughput**, **light grasp**, **light-gathering power**, **optical extent**,^{[1]} and the **AΩ product**. *Throughput* and *AΩ product* are especially used in radiometry and radiative transfer where it is related to the view factor (or shape factor). It is a central concept in nonimaging optics.^{[2]}^{[3]}^{[4]}

From the source point of view, etendue is the product of the area of the source and the solid angle that the system's entrance pupil subtends as seen from the source. Equivalently, from the system point of view, the etendue equals the area of the entrance pupil times the solid angle the source subtends as seen from the pupil. These definitions must be applied for infinitesimally small "elements" of area and solid angle, which must then be summed over both the source and the diaphragm as shown below. Etendue may be considered to be a volume in phase space.

Etendue never decreases in any optical system where optical power is conserved.^{[5]} A perfect optical system produces an image with the same etendue as the source. The etendue is related to the Lagrange invariant and the optical invariant, which share the property of being constant in an ideal optical system. The radiance of an optical system is equal to the derivative of the radiant flux with respect to the etendue.

An infinitesimal surface element, dS, with normal **n**_{S} is immersed in a medium of refractive index *n*. The surface is crossed by (or emits) light confined to a solid angle, d*Ω*, at an angle *θ* with the normal **n**_{S}. The area of d*S* projected in the direction of the light propagation is d*S* cos *θ*. The etendue of this light crossing dS is defined as

As shown below, etendue is conserved as light travels through free space and at refractions or reflections. It is then also conserved as light travels through optical systems where it undergoes perfect reflections or refractions. However, if light was to hit, say, a diffuser, its solid angle would increase, increasing the etendue. Etendue can then remain constant or it can increase as light propagates through an optic, but it cannot decrease. This is a direct result of increasing entropy, which only can be reverted if a priori knowledge is used to reconstruct a phase-matched wave-front such as with phase conjugated mirrors.

Conservation of etendue can be derived in different contexts, such as from optical first principles, from Hamiltonian optics or from the second law of thermodynamics.^{[2]}

Consider a light source *Σ*, and a light detector *S*, both of which are extended surfaces (rather than differential elements), and which are separated by a medium of refractive index *n* that is perfectly transparent (shown). To compute the etendue of the system, one must consider the contribution of each point on the surface of the light source as they cast rays to each point on the receiver.^{[9]}

According to the definition above, the etendue of the light crossing d*Σ* towards d*S* is given by:

The etendue of the whole system is then:

If both surfaces d*Σ* and d*S* are immersed in air (or in vacuum), *n* = 1 and the expression above for the etendue may be written as

The conservation of etendue in free space is related to the reciprocity theorem for view factors.

The conservation of etendue discussed above applies to the case of light propagation in free space, or more generally, in a medium in which the refractive index is constant. However, etendue is also conserved in refractions and reflections.^{[2]} Figure "etendue in refraction" shows an infinitesimal surface d*S* on the *xy* plane separating two media of refractive indices *n*_{Σ} and *n*_{S}.

The normal to d*S* points in the direction of the *z* axis. Incoming light is confined to a solid angle d*Ω*_{Σ} and reaches d*S* at an angle *θ*_{Σ} to its normal. Refracted light is confined to a solid angle d*Ω*_{S} and leaves d*S* at an angle *θ*_{S} to its normal. The directions of the incoming and refracted light are contained in a plane making an angle *φ* to the *x* axis, defining these directions in a spherical coordinate system. With these definitions, Snell's law of refraction can be written as

Radiance of a surface is related to étendue by:

- is the radiant flux emitted, reflected, transmitted or received;
*n*is the refractive index in which that surface is immersed;*G*is the étendue of the light beam.

As the light travels through an ideal optical system, both the étendue and the radiant flux are conserved. Therefore, *basic radiance* defined as:^{[10]}

In the context of Hamiltonian optics, at a point in space, a light ray may be completely defined by a point **r** = (*x*, *y*, *z*), a unit Euclidean vector **v** = (cos *α*_{X}, cos *α*_{Y}, cos *α*_{Z}) indicating its direction and the refractive index *n* at point **r**. The optical momentum of the ray at that point is defined by

In a spherical coordinate system **p** may be written as

Consider an infinitesimal surface d*S*, immersed in a medium of refractive index *n* crossed by (or emitting) light inside a cone of angle *α*. The etendue of this light is given by

Noting that *n* sin *α* is the numerical aperture *NA*, of the beam of light, this can also be expressed as

Note that d*Ω* is expressed in a spherical coordinate system. Now, if a large surface *S* is crossed by (or emits) light also confined to a cone of angle *α*, the etendue of the light crossing *S* is

The limit on maximum concentration (shown) is an optic with an entrance aperture *S*, in air (*n*_{i} = 1) collecting light within a solid angle of angle 2*α* (its acceptance angle) and sending it to a smaller area receiver *Σ* immersed in a medium of refractive index *n*, whose points are illuminated within a solid angle of angle 2*β*. From the above expression, the etendue of the incoming light is

Conservation of etendue *G*_{i} = *G*_{r} then gives

In the case that the incident index is not unity, we have

If the optic were a collimator instead of a concentrator, the light direction is reversed and conservation of etendue gives us the minimum aperture, *S*, for a given output full angle 2*α*.

**^**"17-21-048".*CIE*. Retrieved 2022-02-19.- ^
^{a}^{b}^{c}^{d}^{e}Chaves, Julio (2015).*Introduction to Nonimaging Optics, Second Edition*. CRC Press. ISBN 978-1482206739. - ^
^{a}^{b}Roland Winston et al.,,*Nonimaging Optics*, Academic Press, 2004 ISBN 978-0127597515 **^**Matthew S. Brennesholtz, Edward H. Stupp,*Projection Displays*, John Wiley & Sons Ltd, 2008 ISBN 978-0470518038**^**Lecture notes on Radiance- ^
^{a}^{b}"CIE S 017:2020 ILV: International Lighting Vocabulary, 2nd edition, entry 17-25-077".*cie.co.at/e-ilv*. International Commission on Illumination. Retrieved September 30, 2021. - ^
^{a}^{b}"International Electrotechnical Vocabulary (IEV) Online: Electropedia, IEV ref 845-21-048".*electropedia.org*. International Electrotechnical Commission. Retrieved October 2, 2021. **^***The International System of Units (SI) Brochure, 9th Edition*. International Bureau of Weights and Measures. 2019. ISBN 978-92-822-2272-0.**^***Wikilivre de Photographie*,*Notion d'étendue géométrique*(in French). Accessed 27 Jan 2009.**^**William Ross McCluney,*Introduction to Radiometry and Photometry*, Artech House, Boston, MA, 1994 ISBN 978-0890066782

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- Greivenkamp, John E. (2004).
*Field Guide to Geometrical Optics*. SPIE Field Guides vol.**FG01**. SPIE. ISBN 0-8194-5294-7. - Xutao Sun
*et al.*, 2006, "Etendue analysis and measurement of light source with elliptical reflector",*Displays*(27), 56–61. - Randall Munroe explains why it's impossible to light a fire with concentrated moonlight using a etendue-conservation argument. Munroe, Randall. "Fire from Moonlight".
*What If?*. Retrieved 28 July 2020.